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Theorem cfss 9487
Description: There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfss.1 𝐴 ∈ V
Assertion
Ref Expression
cfss (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem cfss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cfss.1 . . . . . 6 𝐴 ∈ V
21cflim3 9484 . . . . 5 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
3 fvex 6514 . . . . . . 7 (card‘𝑥) ∈ V
43dfiin2 4830 . . . . . 6 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
5 cardon 9169 . . . . . . . . . 10 (card‘𝑥) ∈ On
6 eleq1 2853 . . . . . . . . . 10 (𝑦 = (card‘𝑥) → (𝑦 ∈ On ↔ (card‘𝑥) ∈ On))
75, 6mpbiri 250 . . . . . . . . 9 (𝑦 = (card‘𝑥) → 𝑦 ∈ On)
87rexlimivw 3227 . . . . . . . 8 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) → 𝑦 ∈ On)
98abssi 3938 . . . . . . 7 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On
10 limuni 6091 . . . . . . . . . . . 12 (Lim 𝐴𝐴 = 𝐴)
1110eqcomd 2784 . . . . . . . . . . 11 (Lim 𝐴 𝐴 = 𝐴)
12 fveq2 6501 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1312eqcomd 2784 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (card‘𝐴) = (card‘𝑥))
1413biantrud 524 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥))))
15 unieq 4721 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 𝑥 = 𝐴)
1615eqeq1d 2780 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → ( 𝑥 = 𝐴 𝐴 = 𝐴))
171pwid 4439 . . . . . . . . . . . . . . . . 17 𝐴 ∈ 𝒫 𝐴
18 eleq1 2853 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐴𝐴 ∈ 𝒫 𝐴))
1917, 18mpbiri 250 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴)
2019biantrurd 525 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → ( 𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴)))
2116, 20bitr3d 273 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → ( 𝐴 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴)))
2221anbi1d 620 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → (( 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥))))
2314, 22bitr2d 272 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)) ↔ 𝐴 = 𝐴))
241, 23spcev 3525 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
2511, 24syl 17 . . . . . . . . . 10 (Lim 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
26 df-rex 3094 . . . . . . . . . . 11 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)))
27 rabid 3317 . . . . . . . . . . . . 13 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
2827anbi1i 614 . . . . . . . . . . . 12 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
2928exbii 1810 . . . . . . . . . . 11 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
3026, 29bitri 267 . . . . . . . . . 10 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
3125, 30sylibr 226 . . . . . . . . 9 (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥))
32 fvex 6514 . . . . . . . . . 10 (card‘𝐴) ∈ V
33 eqeq1 2782 . . . . . . . . . . 11 (𝑦 = (card‘𝐴) → (𝑦 = (card‘𝑥) ↔ (card‘𝐴) = (card‘𝑥)))
3433rexbidv 3242 . . . . . . . . . 10 (𝑦 = (card‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥)))
3532, 34spcev 3525 . . . . . . . . 9 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) → ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
3631, 35syl 17 . . . . . . . 8 (Lim 𝐴 → ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
37 abn0 4224 . . . . . . . 8 ({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
3836, 37sylibr 226 . . . . . . 7 (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅)
39 onint 7328 . . . . . . 7 (({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅) → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
409, 38, 39sylancr 578 . . . . . 6 (Lim 𝐴 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
414, 40syl5eqel 2870 . . . . 5 (Lim 𝐴 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
422, 41eqeltrd 2866 . . . 4 (Lim 𝐴 → (cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
43 fvex 6514 . . . . 5 (cf‘𝐴) ∈ V
44 eqeq1 2782 . . . . . 6 (𝑦 = (cf‘𝐴) → (𝑦 = (card‘𝑥) ↔ (cf‘𝐴) = (card‘𝑥)))
4544rexbidv 3242 . . . . 5 (𝑦 = (cf‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥)))
4643, 45elab 3582 . . . 4 ((cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
4742, 46sylib 210 . . 3 (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
48 df-rex 3094 . . 3 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
4947, 48sylib 210 . 2 (Lim 𝐴 → ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
50 simprl 758 . . . . . . . 8 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴})
5150, 27sylib 210 . . . . . . 7 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
5251simpld 487 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ 𝒫 𝐴)
5352elpwid 4435 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥𝐴)
54 simpl 475 . . . . . . 7 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → Lim 𝐴)
55 vex 3418 . . . . . . . . . 10 𝑥 ∈ V
56 limord 6090 . . . . . . . . . . . 12 (Lim 𝐴 → Ord 𝐴)
57 ordsson 7322 . . . . . . . . . . . 12 (Ord 𝐴𝐴 ⊆ On)
5856, 57syl 17 . . . . . . . . . . 11 (Lim 𝐴𝐴 ⊆ On)
59 sstr 3868 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
6058, 59sylan2 583 . . . . . . . . . 10 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ⊆ On)
61 onssnum 9262 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑥 ⊆ On) → 𝑥 ∈ dom card)
6255, 60, 61sylancr 578 . . . . . . . . 9 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ∈ dom card)
63 cardid2 9178 . . . . . . . . 9 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
6462, 63syl 17 . . . . . . . 8 ((𝑥𝐴 ∧ Lim 𝐴) → (card‘𝑥) ≈ 𝑥)
6564ensymd 8359 . . . . . . 7 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ≈ (card‘𝑥))
6653, 54, 65syl2anc 576 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (card‘𝑥))
67 simprr 760 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (cf‘𝐴) = (card‘𝑥))
6866, 67breqtrrd 4958 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (cf‘𝐴))
6951simprd 488 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 = 𝐴)
7053, 68, 693jca 1108 . . . 4 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
7170ex 405 . . 3 (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → (𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴)))
7271eximdv 1876 . 2 (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴)))
7349, 72mpd 15 1 (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wex 1742  wcel 2050  {cab 2758  wne 2967  wrex 3089  {crab 3092  Vcvv 3415  wss 3831  c0 4180  𝒫 cpw 4423   cuni 4713   cint 4750   ciin 4794   class class class wbr 4930  dom cdm 5408  Ord word 6030  Oncon0 6031  Lim wlim 6032  cfv 6190  cen 8305  cardccrd 9160  cfccf 9162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-iin 4796  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-se 5368  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-isom 6199  df-riota 6939  df-wrecs 7752  df-recs 7814  df-er 8091  df-en 8309  df-dom 8310  df-card 9164  df-cf 9166
This theorem is referenced by: (None)
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